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A version of smooth K-theory adapted to the total Chern class

Published online by Cambridge University Press:  18 October 2010

Alain Berthomieu
Alain Berthomieu
Affiliation:
Université de Toulouse, C.U.F.R. J.-F. Champollion, and I.M.T. (Institut de Mathématiques de Toulouse UMR CNRS n° 5219), Campus d'Albi, Place de Verdun, 81012 Albi Cedex, France. alain.berthomieu@univ-jfc.fr
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Abstract

A new model of smooth K0-theory ([5] [1]) is constructed, with the help of the total Chern class (contrary to the theories considered in ]1], [5], [12] and [13] which use the Chern character). The correspondence with the earlier model [1] is obtained by adapting, at the level of transgression forms, the usual formulae which express the Chern character in terms of the Chern classes and vice versa.

The advantage of this new model is that it allows constructing Chern classes with values in integral Chern-Simons characters in a natural way: this construction answers a question asked by U. Bunke [4].

Keywords

multiplicative K-theorysmooth K-theoryChern-Simons transgression
Type
Research Article
Information
Journal of K-Theory , Volume 6 , Issue 2 , October 2010 , pp. 197 - 230
Copyright
Copyright © ISOPP 2010

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References

1.Berthomieu, A.: Direct images for some secondary K-theories, in “From Probability to Geometry (I). Volume in honor of the 60th birthday of Jean-Michel Bismut” (X. Dai, R. Léandre, X. Ma and W. Zhang editors)” Astérisque 327 (2009), 289–360, earlier diffused as three preprints named Direct image for relative and multiplicative K-theories from transgression of the families index theorem, parts 1., 2. and 3., at arXiv:math.DG/0611281, arXiv:math.DG/0703916 and arXiv:0804/0728.Google Scholar
2.Berthomieu, A.: Proof of Nadel's conjecture and direct image for relative K-theory, Bull. Soc. Math. France 130 (2) (2002), 253–307.CrossRefGoogle Scholar
3.Brylinski, J.-L.: Comparison of the Beilinson-Chern classes with the Chern-Cheeger-Simons classes in Advances in Geometry, Progr. Math. 172, Birkhäuser Boston, Boston, MA (1999), 95–105.Google Scholar
4.Bunke, U.: question in the problem session, Oberwolfach Reports 3, Nr 1 (2006), 797.Google Scholar
5.Bunke, U., Schick, Th.: Smooth K-theory, in “From Probability to Geometry (II). Volume in honor of the 60th birthday of Jean-Michel Bismut” (X. Dai, R. Léandre, X. Ma and W. Zhang editors)“ Astérisque 328 (2009), 45–135.Google Scholar
6.Cheeger, J., Simons, J.: Differential characters and geometric invariants, Lecture Notes in Math. 1167, 50–80, Springer, New York, 1985.Google Scholar
7.Felisatti, M.: Differential characters and multiplicative cohomology, K-theory 18 (1999), 267–276.CrossRefGoogle Scholar
8.Freed, D.: Dirac charge quantization and generalized differential cohomology, in Surv. Differ. Geom., VII, 129–194. Int. Press, Sommerville, MA, 2000.Google Scholar
9.Freed, D., Hopkins, M.: On Ramond-Ramond fields and K-theory, J. High Energy Phys. 5, paper 44, 14, (2000).Google Scholar
10.Grothendieck, A.: La théorie des classes de Chern, Bull. Soc. Math. France 86 (1958), 137–154.CrossRefGoogle Scholar
11.Hopkins, M. J., Singer, I. M.: Quadratic functions in geometry, topology and M-theory, J. of Diff. Geom. 70 (3) (2005), 329–452.Google Scholar
12.Karoubi, M.: Théorie générale des classes caractéristiques secondaires, K-theory 4 (1990), 55–87.CrossRefGoogle Scholar
13.Karoubi, M.: Classes caractéristiques de fibrés feuilletés, holomorphes ou algébriques, K-theory 8 (1994), 153–211.CrossRefGoogle Scholar
14.Lott, J.: ℝ/ℤ index theory, Comm. in Anal. and Geom. 2 (1994), 279–311.CrossRefGoogle Scholar
15.Narasimhan, M. S., Ramanan, S.: Existence of universal connections, Amer. J. of Math. 83 (1961), 563–572.CrossRefGoogle Scholar
16.Narasimhan, M. S., Ramanan, S.: Existence of universal connections II, Amer. J. of Math. 85 (1963), 223–231.CrossRefGoogle Scholar
17.Zucker, S.: The Cheeger-Simons invariant as a Chern class, in Algebraic analysis, geometry and number theory (Baltimore, MD, 1988), John Hopkins Univ. Press, Baltimore, MD (1989), 397–417.Google Scholar